Question

If $u={\tan }^{-1}(x+y)$ then $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=$

• A. $\sin 2u$
• B. $\left(1/2\right)\sin 2u$
• C. $2tanu$
• D. ${\sec }^{2}u$

Answer 1

The correct option is B $\left(1/2\right)\sin 2u$

$\tan u=x+y$

$\tan u$ is a homogeneous of degree$=1$

$\sum \frac{xdtanu}{dx}=n\tan u$

$se{c}^{2}u\sum \frac{xdu}{dx}=\tan u$

$\sum \frac{xdy}{dx}=\frac{1}{2}\sin 2u$